College Algebra, Chapter 4, 4.5, Section 4.5, Problem 20

Factor the polynomial $P(x) = x^2 - 8x + 17$, and find all its zeros. State the multiplicity of each zero.

To find the zeros of $P$, we set $x^2 - 8x + 17 = 0$, by using quadratic formula


$\begin{equation} \begin{aligned} x =& \frac{-(-8) \pm \sqrt{(-8)^2 -4(1)(17)}}{2(1)} \\ \\ =& \frac{8 \pm \sqrt{-4}}{2} \\ \\ =& \frac{8 \pm 2 \sqrt{-1}}{2} \\ \\ =& 4 \pm \sqrt{-1} \\ \\ =& 4 \pm i \end{aligned} \end{equation}$


By factorization,


$\begin{equation} \begin{aligned} P(x) =& x^2 - 8x + 17 \\ \\ =& \left[ x - (4 + i) \right] \left[ x - (4 - i) \right] \end{aligned} \end{equation}$


Therefore, the multiplicity of each zero is $1$.